Can you find what to invoke?

For discussing Olympiad Level Number Theory problems
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Masum
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Can you find what to invoke?

Unread post by Masum » Wed Dec 17, 2014 7:33 pm

This is a really nice problem. If the solution just crosses your mind, great! If not, keep thinking. I can assure you, when you find the solution you will feel real good.
Prove that there are an infinite integers that can't be written as $x^{\tau(x)}+y^{\tau(y)}$.
Here, $\tau(n)$ is the number of positive divisors of $n$, as usual.
One one thing is neutral in the universe, that is $0$.

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Fm Jakaria
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Re: Can you find what to invoke?

Unread post by Fm Jakaria » Wed Dec 17, 2014 8:48 pm

Any number of the form $4x+3$ works. This is because: $n^{\tau(n)}$ is a perfect square for all positive integers n.[If n is itself perfect square, done. Else $\tau(n)$ is even, and done.] And $x^2+y^2$ is 0 or 1 or 2 mod 4.

Note: I am not feeling good....
You cannot say if I fail to recite-
the umpteenth digit of PI,
Whether I'll live - or
whether I may, drown in tub and die.

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Masum
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Location:Dhaka,Bangladesh

Re: Can you find what to invoke?

Unread post by Masum » Thu Dec 18, 2014 2:41 pm

" If the solution just crosses your mind, great!" did you miss that part? Because the latter part was kind of for newbies
One one thing is neutral in the universe, that is $0$.

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